KANSAS CITY, Mo. — Buddy Biancalana still has to talk about David Letterman. It is both his blessing and curse. He has been telling the story for 33 years, of course, to reporters, fans, and pretty much anyone who asks. And believe it or not, this happens more often than you would think.

“It all started in August,” Biancalana says.

Biancalana, 58, is talking about the summer of 1985, which, in fact, is why he’s standing here in a front lobby at Kauffman Stadium. It’s Saturday evening, and the Royals are celebrating their 1985 World Series team before a game against the St. Louis Cardinals. Biancalana, a former shortstop, is in town to see old teammates and toast the past. He’s also talking about Letterman, the former late-night king, with a smile on his face.

Well, as much of a smile you can muster when you’ve told this story for 33 years. To a certain generation of Royals fans, the story needs little introduction. In the spring of 1985,...